external disturbances and coupling mechanisms in...

Proceedings of the ASME 2010 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC/CIE 2010August 15-18, 2010, Montreal, Quebec, Canada

DETC2010-29004

EXTERNAL DISTURBANCES AND COUPLING MECHANISMSIN UNDERACTUATED HANDS

Ravi Balasubramanian∗, Joseph T. Belter, and Aaron M. Dollar

Department of Mechanical Engineering

Yale University

New Haven, Connecticut 06520.

{ravi.balasubramanian, joseph.belter, aaron.dollar}@yale.edu

ABSTRACT

With the goal of improving the performance of under-

actuated robotic hands in grasping, we investigate the influ-

ence of the underlying coupling mechanism on the robust-

ness of underactuated hands to external disturbance. This

paper identifies unique behaviors in the hand’s response

as a function of the coupling mechanism and the actua-

tion mode the hand is operated in. Specifically, we show

that in conditions when the actuator position is fixed, hands

with single-acting mechanisms exhibit a bimodal behavior

in contrast to hands with double-acting mechanisms that

exhibit a unimodal behavior. We then present an analysis

of how these behaviors influence grasping capability of the

hand and then discuss implications for underactuated hand

design and operation.

1 INTRODUCTION

There has long been a desire to minimize the number of

actuators in robotic and prosthetic hands due to constraints

on size and mass. This has been accomplished by cou-

pling the motion of joints, with many designs having fewer

actuators than degrees of freedom. Such robotic hands,

termed “underactuated”, also exhibit passive adaptability

between joints and digits; that is, in certain situations, these

hands naturally change posture to adapt to the environment.

While the performance and design of underactuated hands

when driven through internal actuation have been analyzed

in prior work [1–5], the behavior of these hands when they

∗Contact author

are driven by external forces is still not well understood.

This “disturbance response” behavior is the subject of this

paper.

Several clever underactuated hand designs have been

proposed in prior work. While the hands may differ in

the number of fingers and the number of links in each

finger, this paper focuses on the differences between the

hands in terms of the coupling transmission design and the

actuation mode for flexion (curling) and extension (open-

ing) finger motion. There are two primary types of actuat-

ing mechanisms: single-acting and double-acting. Single-

acting mechanisms control only one of either the flexion

or extension of the finger (see Figs. 1a and 1c). This is

achieved using an actuator that can only pull (for example,

a single cable routing) or push (for example, a plunger), and

the reverse motion is achieved using springs. Examples of

robotic hands with single-acting mechanisms include the

SDM [1], Balance Bar [6], and 100G robotic hands [7].

In contrast, double-acting mechanisms control both the

flexion and extension motion of the fingers (see Fig 1d).

This is achieved through different methods, such as four-

bar linkages, double cable routing, or gears. Note that

springs may still be used to ensure mechanism stability and

compliance. Examples of robotic hands with double acting

mechanisms include the Laval Hands [8, 9], SPRING [10],

Southampton [11], Graspar [12], BarrettHand [13], and

Obrero [14] robotic hands. In addition, some double-acting

hands such as the BarrettHand and the Southampton hand

feature clutch mechanisms that engage or disengage the

1 Copyright c© 2010 by ASME

(a) (b) (c)

(d)

Fa

Fa

Fa

(e)

τa

Figure 1. EXAMPLES OF UNDERACTUATED HANDS: (A) SINGLE

ACTING CABLE-DRIVEN SYSTEM, (B) SINGLE-ACTING WHEN COU-

PLING BREAKS DOWN (CABLE SLACK), (C) SINGLE-ACTING WITH

EXTRA DEGREE OF FREEDOM (PLUNGER CAN PUSH, BUT NOT

PULL), (D) DOUBLE-ACTING LINKAGE-DRIVEN SYSTEM, (E) CLUTCH

MECHANISM SYSTEM

joint coupling based on actuation conditions (see Fig. 1e).

The manner in which the underactuated hands reshape

in the presence of external forces is important. External

forces can arise in several situations, including unplanned

collisions, vibration of the base, or changing force from

another finger. These events can occur when the hand

approaches an object or when the hand is already closed

on an object. Forces acting on the phalanxes are trans-

ferred through the mechanism all the way back to the actu-

ator. As a result, the effect of these forces is a function of

not only the coupling scheme in the mechanism, but also

the nature of the actuation, including backdrivability and

control mode (for example, force control or position con-

trol). While the equilibrium of underactuated grasping in

the presence of disturbances has been studied before [15],

the influence of the coupling mechanisms on the hand’s re-

sponse is still not clearly understood. A certain disturbance

force might result in grasp failure in one coupling and ac-

tuation scheme, particularly when are a large number de-

grees of freedom, and in a more stable grasp in another

situation. It is therefore important to analyze how various

configurations behave in response to disturbances and use

θ2

θ1

f1

f2

k1

k2

Proximal link

(length l1)

Distal link

(length l2)

X

Y

Figure 2. TWO-LINK REVOLUTE-REVOLUTE FINGER. THE ACTU-

ATION CAN BE EITHER A SINGLE-ACTING OR DOUBLE-ACTING

MECHANISM (SEE FIG. 1)

this understanding to design the most appropriate coupling

and actuation schemes.

In this paper, we first present a framework for studying

the response of underactuated hands to external disturbance

forces, taking into consideration the kinematics of joint

coupling, external forces, and control modes (section 2).

Then in section 3, we present results from an analysis of

the configuration change of two-link underactuated fingers,

incorporating cable-driven systems and linkage driven sys-

tems, in response to external forces. We also present how

these configuration changes potentially affect a grasp. Fi-

nally, in section 4, we present a discussion of the interesting

behaviors arising from the combination of control modes

and the coupling mechanism and how this analysis can in-

form the design and operation of underactuated hands.

2 FRAMEWORK FOR UNDERACTUATED HAND

ANALYSIS

Our framework for analyzing underactuated hands con-

sists of three components: 1) The static equilibrium equa-

tions that relate contact forces on the phalanxes with joint

torques; 2) The kinematic coupling between the phalanxes

and the actuator; 3) The change in robot configuration due

to the external forces in the presence of joint coupling. As

an example, we consider the flexion and extension behavior

of a two-link revolute-revolute finger with a single actua-

tor (see Fig. 2) .

2.1 Static Equilibrium

The relationship between the contact forces and the net

joint torques in the finger (see Fig. 2) may be expressed as

τ = JTc fe, (1)


where fe =

(

f1

f2

)

represents the normal contact force on

the proximal and distal links, τ =

(

τ1

τ2

)

the resulting

torque at the joints, and Jc ∈ R2×2 the Jacobian that maps

between the two spaces. For a two-link mechanism, the

Jacobian Jc can be computed as [2]

Jc =

(

k1 0

k2 + l1 cosθ2 k2

)

, (2)

where k1 represents the proximal force location, k2 the

distal force location, l1 the proximal link length, l2 the dis-

tal link length, θ2 the relative angle between the two links.

While this formulation assumes that the contact point can

slide on the link without friction, friction models can easily

be incorporated into this framework. We present a brief dis-

cussion of the influence of the contact types on this analysis

in section 4.

2.2 Kinematics of the Coupling Mechanism

The kinematics of the coupling mechanism may be ex-

pressed as a first-order differential equation in the mecha-

nism’s configuration θ =

(

θ1

θ2

)

and actuator variable θa.

In cable-driven mechanisms, the actuator variable may be

defined as the angle traveled by the actuator pulley over

which the cable travels, while in linkage driven mecha-

nisms, the actuator variable may be defined as the angle

traveled by the actuating link.

For cable driven mechanisms in current underactuated

hands such as the SDM hand, the kinematics of the cou-

pling mechanism may be expressed as

∆θa = r1∆θ1 + r2∆θ2, (3)

where r1 and r2 represent the pulley radii (assuming unit

radius for the actuation pulley).

For four-bar linkage driven mechanisms in current un-

deractuated hands such as the SARAH hand, the kinematics

of the coupling mechanism may be expressed as

∆θa = ∆θ1 +∆θ2R, (4)

where R represents the transmission ratio of the mecha-

nism. Note that for a four-bar linkage mechanism, the

transmission ratio R is a function of joint configuration θ

and link lengths. In this paper, we consider only small joint

configuration changes from a given configuration θ. Thus,

R is treated as a constant for the instantaneous analysis in

this paper.

A closer analysis of (3) and (4) shows that the kine-

matics of both cable-driven mechanisms and linkage-driven

mechanisms can be expressed as

Ja∆θ = ∆θa, (5)

where Ja =(

a1 a2

)

represents the actuator Jacobian of the

mechanism. For linkage-driven systems, a1 equals 1 and

a2 equals R. For cable-driven systems, a1 equals r1 and a2

equals r2. For both types of systems, the transmission ra-

tio may be defined as R = a2/a1. This common structure

between the coupling mechanisms of existing underactua-

tion hands induces the hands to behave in a similar fash-

ion when driven by the actuator [2]. However, as will be

shown, these mechanisms behave differently in the pres-

ence of disturbance forces.

Some double-acting hands like the BarrettHand [13]

and the Southampton hand [11] use either clutch or brake

mechanisms to enable a multimodal joint coupling. For ex-

ample, the BarrettHand switches the joint coupling depend-

ing on the proximal joint torque value, while the Southamp-

ton hand uses brakes to specify the joint coupling. The joint

coupling for the BarrettHand may be expressed as:

∆θ2 = α∆θ2, if τ1 < τmax

∆θ1 = 0, ∆θ2 > 0, if τ1 > τmax

}

, (6)

where τmax represents the maximum allowed torque at the

proximal joint and α the fixed non-adaptive coupling ra-

tio between the two joints. Note that the BarrettHand is

not backdrivable and is not compliant because of a double-

acting worm gear mechanism. Similarly, the Southamp-

ton hand uses a rigid coupling mechanism and is non-

backdrivable because of the lead-screws that drive the

joints. Thus, these hands are rigid to external disturbances.

We will not explore such hands with such rigid coupling

mechanisms further in this paper.

2.3 Robot Configuration Change Due to External

Force

An external disturbance force fe on the robot can cause

a change in configuration ∆θ. The magnitude and direction

of ∆θ in the joint configuration space depends on factors

such as 1) the joint coupling, 2) the direction, magnitude,

and location of the disturbance force, 3) the hand control

mode, and 4) joint compliance. This paper explores the

change in robot configuration as a function of these factors,

specifically exploring how ∆θ varies with force direction

(tending to either flex or extend the link), magnitude, and


point of application. We also look at two different con-

trol modes for the actuator: force control and position con-

trol. To reduce the size of the parameter space, we assume

disturbance forces are applied normal to the links (that is,

frictionless contacts), and do not vary joint stiffness. We

also model external forces as constant, but models where

the external force changes with deflection (such as com-

pliant springy contacts) and/or models including geometric

constraints imposed by external contacts can also be incor-

porated using the same framework (albeit with differing re-

sultant behaviors). The contact forces are intentionally kept

simple in this paper in order to focus on the hand configu-

ration change.

The configuration change ∆θ for an external force fe

can be quantified using a Lagrangian view of the work done

by the external forces and the energy stored in the springs in

the presence of the actuation constraints [16]. Specifically,

we can define the Lagrangian L as

L = ∆Ws +∆Wc +∆Wa, (7)

where Ws represents the work done on the springs, Wc the

work done by the external forces, Wa the work done on the

actuator.

The work done on the spring ∆Ws = −1/2∆θT KJ∆θ

and work done by the external forces ∆Wc = f Te Jc∆θ [17]

are similar in form for all of the underactuated mechanisms

we consider. Here KJ =

(

KJ1 0

0 KJ2

)

represents joint stiff-

ness. The work done on the actuator Wa, however, takes dif-

ferent forms depending on the control mode the mechanism

is driven in (see Table 1). In the force-control mode (that

is, actuator force is controlled to be constant while actuator

position can vary), the work is “real”, while in the position

control mode (that is, actuator position is held fixed and

force can vary), the “virtual” work must equal zero [18]. In

the work equation for the position control mode, p is the

pretension in the actuating mechanism that ensures mecha-

nism stability prior to the application of the external distur-

bance fe.

By taking derivatives of the Lagrangian with respect to

the variables and any Lagrange multipliers, we can derive

the static balance equations [16]. Table 1 presents the static

balance equations for the two-link mechanism, one for each

control mode. Note that these equations predict the instan-

taneous small changes in hand posture ∆θ from a statically

stable configuration as a result of the external force and the

actuation mode.

A closer look at the static equations reveals that the sys-

tem response in the decoupled mode (that is, actuator ap-

plies no load or position constraint, such as the case when

a tendon goes slack) is shaped primarily by the joint stiff-

nesses KJ1 and KJ2. The system response in the force con-

trol mode is shaped by the joint stiffnesses KJ1 and KJ2,

the constant actuation force fa, and pulley radii r1 and r2.

Thus, the system response in the force control mode may

be viewed as a modified version of the system response in

the decoupled mode, since the only difference is work done

on the actuator. In contrast, the system response in the po-

sition control mode is shaped by the joint stiffnesses KJ1

and KJ2 and the pulley radii ratio R (due to the kinematic

constraint created by constant cable length), and the actua-

tor pretension p.

3 RESULTS

By using the SDM hand as an exemplar of a hand with

a single-acting mechanism and the SARAH hand as an ex-

emplar of a hand with a double-acting mechanism, we now

present results from the simulation of the models presented

in section 2.3 to compute hand configuration change for

external loads. We also include an analysis of the potential

impact of such finger deviations on a grasp by analyzing

the kinematics of a two-link mechanism. In the interest of

keeping the parameter space small, we assumed that there

was a force only on the distal joint (that is, f1 = 0), but the

core of the framework used for analysis holds for distur-

bance forces on the proximal link also. Table 2 shows the

other parameters used in the analysis.

3.1 Response Variation As a Function of Contact

Force Location and Magnitude

The models in section 2.3 permit a quantification of

configuration change for different external loads. The re-

sults for the double-acting (such as a linkage-driven mecha-

nism) and the single-acting actuation case (such as a cable-

driven mechanism) for the decoupled and force-control

modes are shown in Figs. 3a and 3b. The results for the

double-acting actuation case with position-control mode is

shown in Fig. 3c. These contour plots show the change

in distal joint configuration ∆θ2 of a two-link mechanism

as a function of the magnitude and direction of contact

force (horizontal axis) and the force contact location (verti-

cal axis). Note that similar contour plots can be derived for

the change in proximal joint configuration ∆θ1.

As expected, Fig. 3a shows that for the decoupled


Table 1. EFFECT OF CONTROL MODE ON WORK DONE ON ACTUATOR BY EXTERNAL FORCES

Actuation mode Work done on

actuator

Static equations Example scenario

Force control ∆Wa = fa∆θa1 −K∆θ+ JT

c fe + JTa fa = 0 (8) Maintaining fixed cable ten-

sion in SDM hand.

Position control Virtual work

∆Wa =(λ− p)∆θa = 0 2

Ja∆θ = 0

−K∆θ+ JTc fe + JT

a (λ− p) = 0

(9) 1) Maintaining fixed cable

length in SDM hand; 2) Non-

backdrivability in SARAH

hand.

Decoupled (see

Fig. 1b)

Wa = 0 −K∆θ+ JTc fe = 0 (10) Cable slackening in SDM

hand.

1: fa is the constant actuation force. 2: λ is the tendon force resulting from the coupling constraint.

-0.15

-0.1

-0.05

-0.05

0 0

0.05

0.1

0.05-0.4

-0.2

0

0.2

0.4

Force

location

-0.48

-0.24

0 0.48

-30 -20 -10 0 10 20 30

Disturbance force f2 (N)

0

0.02

0.04

0.06

0.08

0.10

k2 (m)

0.24

0.15

-30 -20 -10 0 10 20 30


-30 -20 -10 0 10 20 30


(a) (b) (c)

Figure 3. DISTURBANCE RESPONSE OF A TWO-LINK HAND IN (A) DECOUPLED MODE AND (B) FORCE CONTROL MODE. NOTE THAT (A) AND

(B) APPLY FOR BOTH SINGLE-ACTING AND DOUBLE-ACTING MECHANISMS. (C) DISTURBANCE RESPONSE OF A TWO-LINK HAND IN POSITION

CONTROL MODE WITH A DOUBLE-ACTING MECHANISM. THE CONTOURS SHOW VARIATION IN DISTAL JOINT DEVIATION AS A FUNCTION OF

DISTURBANCE FORCE f2 AND ITS LOCATION k2.

mode, the distal joint closes in (∆θ2 positive) for flex-

ion external forces and opens out for extending external

forces (∆θ2 negative). The system response in force con-

trol mode (see Fig. 3b) is similar to the system response in

the decoupled mode, except that the work done on the ac-

tuator slightly “warps” the contours. For the double-acting

mechanism in position control mode (see Fig. 3c), we see

a critical distal force location k2 at which the mechanism

does not move (∆θ2 = 0) for any external force, indicating

that the mechanism is extremely stiff at that force location.

This location has been termed the equilibrium point for the

mechanism in prior work [2, 19].

The results for the single-acting actuation case (such

as a cable-driven mechanism) in position control mode are

shown in Fig. 4. This plot shows the change in distal-

joint configuration change ∆θ2 of a single-acting two-link

mechanism in position control mode with the joint cou-

pling given by (5). The overall response of the system

is shaped by the joint stiffnesses KJ1 and KJ2, the pulley

radii ratio R (due to the kinematic constraint created by

constant cable length), and actuator pretension. The sys-

tem response is essentially a hybrid between the decou-


Table 2. SYSTEM PARAMETERS

Proximal joint stiffness KJ1 1 Nm/rad

Distal joint stiffness KJ2 5 Nm/rad

Proximal and distal link length l1 and l2 0.1 m

Proximal pulley radius r1 0.02 m

Pulley radii ratio R 0.6 m

Proximal joint configuration θ1 π/10 rad

Distal joint configuration θ2 π/3 rad

Cable pretension p 10 N

Cable actuation force fa 10 N

External force on proximal link f1 0 N

-0.028

0.028

0.056

0.056

0.084

0.112

0.14

-0.028

-30 -20 -10 0 10 20 30


0

0.02

0.04

0.06

0.08

0.10

k2 (m)

Force

location

0

0.112

A B

CDE

Figure 4. DISTURBANCE RESPONSE OF AN SINGLE-ACTING TWO-

LINK HAND IN POSITION CONTROL MODE: VARIATION IN DISTAL

JOINT DEVIATION AS A FUNCTION OF DISTURBANCE FORCE f2

AND ITS LOCATION k2.

pled mode (right of the dashed red line) and double-acting

mechanism in position control mode (left of the dashed

line) seen in Figs. 3a and 3c. The dashed (red) line repre-

sents the combination of force magnitude f2 and location k2

at which the inter-joint coupling breaks down (for example,

the cable going slack in the SDM hand). This dashed line

is a function of the tendon tension (p=10 N in this exam-

ple), and would lie on the f2 = 0 line if the pretension p

was zero.

To the left of the dashed line, we have a ∆θ2 = 0 con-

tour line when f2 = 0, and there is an equilibrium point

when ∆θ2 = 0 for any external force, just as in the case

-30 -20 -10 0 10 20 300

0.04

0.08


k2 (m)

Force

location

p= −40 p=40

p=0

Figure 5. CONTACT FORCE AND LOCATION COMBINATIONS THAT

NULLIFY THE PRETENSION p IN THE ACTUATOR.

of position control for a double-acting system (see Fig. 3c).

But there exists a ∆θ2 = 0 contour to the right of the dashed

line also, due to the flexion motion of the distal link for

flexion forces in the absence of joint coupling. The transi-

tion from a (coupled) position control mode to a decoupled

mode produces interesting behavior when either k2 or f2 is

small (region ABCDE in Fig. 4). The distal joint changes

from flexion to extension to flexion in a small force range

with force location 0.04 m < k2 < 0.075 m. Note that a

similar contour plot can be drawn for proximal link devia-

tion ∆θ1 also.

Fig. 5 shows the combination of external force on the

distal link and its location that nullifies a pre-existing actu-

ation force (varied here between p =−40 N and p = 40 N).

Note that pretension can be positive or negative for double-

acting systems, but can be either only positive or only neg-

ative for single-acting systems. The contours in this plot

are similar to the dashed red line in Fig. 4.

3.2 Variation of the Equilibrium Point in Position Con-

trol Mode

As noted in Fig. 3c, there is a critical value of force lo-

cation k2, called the equilibrium point, at which the under-

actuated hand does not move (∆θ = 0) in position control

mode. The equilibrium point e is a function of the mech-

anism design parameters such as proximal link length l1,

the pulley radii ratio R, and distal joint angle θ, and has the

form e = l1Rcosθ2/(1−R). Fig. 6 shows that the equilib-

rium point moves distal rapidly as the pulley radii ratio R

increases toward unity (for fixed distal joint angle θ2 = π/3

). This indicates that the equilibrium point could be beyond

the distal link length. In contrast, the equilibrium point

moves proximal on the distal link as the pulley radii ra-


-0.448

2-0.168-0.112-0.056

0.056 0.112 0.168

0.392

00.05 0.10 0.15 0.20

1.0

1.5

2.0

Proximal link length l1

Pulley

radii

ratio R

0

0.5

Figure 6. VARIATION OF THE EQUILIBRIUM POINT (CONTOURS) AS

A FUNCTION PROXIMAL LINK LENGTH l1 AND THE PULLEY RADII

RATIO R.

tio R decreases toward unity, indicating that the equilibrium

point is below the distal joint.

3.3 Grasping Behaviors Arising from Hand Configu-

ration Change

Fig. 7 shows the joint configuration change space (for

the two-link finger) split into four regions depending on the

direction of potential fingertip movement in the X-direction

and proximal and distal joint configuration change. The

regions represent four behaviors: 1) Squeezing, 2) Caging,

3) Ejection, and 4) Release. Table 3 presents the potential

effect of these behaviors on a grasp. Note that prior work

in grasping has also explored ejection [2, 20] (albeit in a

different form), caging [21], and hold [22] behaviors.

However, a configuration change of an arbitrary direc-

tion is not possible in underactuated hands. When under-

actuated hands are operated in force control mode, then the

direction of proximal or distal joint motion depends pri-

marily on the direction of external force. The direction of

motion of the proximal joint and distal joint would be iden-

tical, indicating a positive slope in the joint configuration

space. The exact magnitude of the slope would depend on

the relative magnitudes of joint compliance. Thus, the hand

would exhibit release behavior for extending forces on the

distal link that overcome the cable tension and squeeze be-

havior for flexion forces on the distal link (see Fig. 8a).

Distal

Joint

∆θ2

(rad)

Proximal Joint ∆θ1 (rad)

0.1 0.2-0.2 -0.1

-0.2

0.2

Squeeze

Release

Cage

Eject

0

-0.1

0.1

0

Figure 7. GRASPING BEHAVIORS OF A TWO-LINK FINGER AS A

FUNCTION OF JOINT DEFLECTION. THE FINGERTIP HAS DISPLACE-

MENT ALONG THE NEGATIVE X DIRECTION IN THE REGION TO THE

RIGHT OF THE RED SOLID LINE.

Table 3. ROBOT HAND BEHAVIORS AND EFFECT ON A GRASP

Behavior Fingertip and joint

motions

Potential effect on

grasp

Squeeze Negative X-motion;

both joints curl in

Enveloping object

with potentially

multiple contacts.

Caging Negative X-motion;

distal joint curls,

proximal joint opens

Enveloping object

with potential distal

contact. Proximal

contact weakens.

Tends to pull object

inward.

Ejection Positive X-motion;

proximal joint curls,

distal joint opens

Potential contact

with proximal

phalanx, but distal

joint moves away

potentially break-

ing contact. Tends

to push object

outward.

Release Positive X-motion;

both joints open

Both phalanxes po-

tentially break con-

tact.


CageEject

Cage Eject

-30 0 30Disturbance force f

2 (N)

SqueezeEject

Cage

Eject0

0.04

0.08

k2 (m)

Force

location

(a) (b)

-30 0 30 -30 0 30

(c)

Release

Squeeze

Squeeze

Release

Cage

Eject

1

2

3

4

5

6

7

(d)

1

2

3,5

4

1 2 3 4 5 1 2 3

4

5 6 7

Double-acting

mechanism Single-acting

mechanism

Proximal Joint ∆θ1 (rad)

Distal

Joint

∆θ2

(rad)

0

Figure 8. RESPONSE BEHAVIOR OF TWO-LINK FINGER IN

A) FORCE CONTROL MODE, B) DOUBLE-ACTING POSITION CON-

TROL MODE, AND C) SINGLE ACTING POSITION CONTROL

MODE TO EXTERNAL DISTURBANCE, D) GRASPING BEHAVIOR OF

THE SINGLE-ACTING AND DOUBLE-ACTING MECHANISMS WHEN

VIEWED IN JOINT CONFIGURATION-CHANGE SPACE. THE NUM-

BERED LOCATIONS CORRESPOND TO A VARIATION IN EXTERNAL

LOAD FROM THE EXTENSION DIRECTION TO THE FLEXION DIREC-

TION INDICATED IN (B) AND (C).

However, we get interesting behaviors when the hand

is operated in position control mode. Figs. 8b and 8c show

the qualitative behavior of single-acting and double-acting

underactuated hands to external disturbances when oper-

ated in position control mode with the parameters in Ta-

ble 2. We notice that double-acting mechanisms exhibit

caging and ejection behaviors only, depending on the dis-

tal force location and magnitude. In contrast, we notice

that the single-acting mechanism exhibits caging, ejection,

and squeezing behaviors, depending on the external force

location and magnitude. Note that these grasping behav-

iors arising from configuration changes are a function of

joint configuration, the coupling mechanism, the number of

degrees of freedom in the finger, and external constraints,

and the system will respond differently as the parameters

change.

4 DISCUSSION

One of the goals of our research is to build underactu-

ated robotic hands that offer near-perfect grasping capabil-

ity in different scenarios. Most previous studies of underac-

tuated hands have explored design issues focused on favor-

able behavior when the hands are actuated internally (force

control mode; see Table 1) [1–5].

Even though we consider only a simple two-link fin-

ger in the absence of external constraints in this paper, a

close analysis of these systems in position control mode re-

veals interesting aspects of their behavior. Note that oper-

ating a robotic hand in position control mode is analogous

to using non-backdriveable transmissions, which is advan-

tageous from a power consumption standpoint. We believe

that outlining the behaviors seen in these examples will of-

fer a better understanding the behavior of underactuated

hands with more degrees of freedom and in the presence

of external constraints.

4.1 Bimodal Response of Single-Acting Underactu-

ated Hands In Position Control Mode

While double-acting mechanisms offer a smooth vari-

ation in response in position control mode (see Fig. 3c),

single-acting mechanisms offer a bimodal response in po-

sition control mode (see Fig. 4). Specifically, single-acting

hands like the SDM hand in position control mode behave

similar to double-acting mechanisms when the cable is taut.

However, a sufficiently large external force in the flexion

direction can cause the cable to go slack, and thus elimi-

nate the coupling between the two joints. In this decoupled

mode, the hand behaves like a passive compliant two-link

mechanism and naturally complies with the external force.

The external load at which the transition from the position

control mode to the decoupled mode occurs is determined

by the pretension in the actuation mechanism (see Figs. 4

and 5).

4.2 Grasping Behaviors In Underactuated Hands

The grasping behaviors of underactuated hands have

been analyzed in several contexts, such as the robustness to

uncertainty in hand position relative to the object [23], the

sliding of the hand on the object [2, 24], inadvertent con-

tact forces [25], and adaptability [3–5]. In section 3.3, we

presented a novel approach to qualifying small hand move-

ments in response to external disturbances in the context

of grasping. Specifically, we use the inward movement of

the fingertip and distal joint flexion to distinguish hand be-


haviors such as squeezing and caging which tend to wrap

the fingers around an object. In contrast, behaviors such

as ejection and release tend to unwrap the fingers from the

object. Note that this approach of using fingertip motion

to qualify a grasping motion depends on the instantaneous

joint configuration and the link lengths.

Using this metric over the space of hand configura-

tion changes, we noticed that current underactuated hands

in position control mode switch between different behav-

iors (good and bad) depending on circumstances. Specifi-

cally, hands with double-acting mechanisms produce only

caging and ejection behavior. Furthermore, they exhibit

ejection behavior even for flexion disturbance forces (see

Fig. 8b and 8d). Alternatively, hands with single-acting

mechanisms like the SDM hand produce caging, ejection,

and squeeze behavior (see Fig. 8c and 8d). The squeeze be-

havior occurs when the joint coupling breaks down and is

good for grasping since the hand complies with the exter-

nal load in a favorable direction. Interestingly, hands with

single-acting mechanisms produce an ejection behavior for

small external loads (region ABCDE in Fig. 4), but tran-

sition into squeeze behavior for larger forces. If the eject

behavior for small loads (region ABCDE) is to be elimi-

nated in cable-driven systems, we infer from Figs. 4 and 5

that the pretension must be zero. In such a condition, any

flexion load will cause the mechanism to enter the squeeze

behavior. Note that a zero pretension in the actuation mech-

anism is not always possible, particularly when the hand is

already applying forces to a grasped object. More work

is required to determine how to get advantageous behav-

ior from underactuated hands in position control mode for

arbitrary external loads, particularly in the presence of ex-

ternal constraints and as the number of degrees of freedom

increases.

4.3 Effect of Equilibrium Point on Grasping Behav-

iors

The equilibrium point has been used in previous work

as the point towards which a precision grasp’s contacts can

slide (when internally actuated) for the grasp to become

stable [2]. Thus, underactuated hands have been designed

to locate the equilibrium point within the distal link, with-

out which the object may be ejected. From our analysis

of the response of the fingers to external disturbance (see

Figs. 8b and 8c), we notice that another aspect of the equi-

librium point may be exploited as well. Both the double-

acting and single-acting mechanisms in position control

mode (see Figs. 8b and 8c) transition between different be-

haviors (caging and ejection in this instance) at the equilib-

rium point. Thus, the finger could be designed to place the

equilibrium point so as to maximize the areas of cage and

squeeze behaviors across the mechanism’s joint configura-

tion space. More work is required to provide a unified view

of underactuated grasping in the light on internal actuation

and external disturbances.

It is also interesting that since joint stiffnesses do not

affect the equilibrium point, they can be chosen based on

other factors. For instance, the ratio of the joint stiffnesses

can be set in order to ensure that the proximal joint closes

in on an object at a faster rate than the outer joint when ac-

tuated to ensure maximum grasping ability in the presence

of uncertainty (as is suggested in [26]).

4.4 Modeling of Contact Forces

The external forces that we have used in this paper have

been intentionally kept simple so that the analysis can focus

on the deflections of the mechanism. Specifically, we have

used contact forces that slide on the links [2] and remain

constant as the mechanism changes configuration. How-

ever, note that the choice of contact force model does not

affect the core of the framework used for analysis. Exter-

nal forces that arise from compliant and sticky contacts can

easily be included in the Lagrangian formulation [16].

While our paper has quantified the configuration

changes of the hand as a function of external loads, we

have only shown qualitatively how such deflections may in-

fluence grasps. Once the kinematic constraints arising from

contacts with objects and the environment are included, our

framework for analyzing hand configuration change will in-

form how the contact forces change and influence object

stability. This will then permit us to leverage prior work in

quantifying grasp stability [2, 27].

ACKNOWLEDGMENT

The authors thank Lael Odhner for interesting discus-

sions on Lagrangian formulation of mechanical systems.

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